MiS Preprint Repository

We study the unextendible maximally entangled bases (UMEB) in ${\mathbb{C}}^d⨂{\mathbb{C}}^d$ and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in ${\mathbb{C}}^d⨂{\mathbb{C}}^d$, there is a partial Hadamard matrix which can not be extended to a complete Hadamard matrix in ${\mathbb{C}}^d$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can be extended to a complete Hadamard matrix, which answers a conjecture about $d=5$. We obtain that for any $d$ there is a UMEB except for $d=p\text{or}2p$, where $p\equiv 3\mathrm{mod}4$ and $p$ is a prime. The existence of different kinds of constructions of UMEBs in ${\mathbb{C}}^{nd}⨂{\mathbb{C}}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times 5\times 7$ is also discussed.